Thermonuclear reaction rate of 56 Ni p, 57 Cu and 57 Cu p, 58 Zn

PHYSICAL REVIEW C, VOLUME 64, 045801 Thermonuclear reaction rate of 56 Ni p, 57 Cu and 57 Cu p, 58 Zn O. Forstner, 1, * H. Herndl, 1 H. Oberhummer, 1 H. Schatz, 2, and B. A. Brown 3 1 Institut für Kernphysik, Technische Universität Wien, Wiedner Hauptstraße 8 10, A-1040 Wien, Austria 2 Gesellschaft für Schwerionenforschung, Planckstraße 1, D-64291 Darmstadt, Germany 3 Department of Physics and Astronomy, and National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, Michigan 48824 Received 25 May 2001; published 30 August 2001 Updated stellar rates for the reactions 56 Ni(p,) 57 Cu and 57 Cu(p,) 58 Zn are calculated by using all available experimental information as well as input data from the nuclear shell model. We calculated Coulomb and Thomas-Ehrmann shifts to obtain the unknown excitation energies and the spin/parity assignments of 58 Zn. The astrophysical consequences are discussed. DOI: 10.1103/PhysRevC.64.045801 PACS numbers: 26.30.k, 21.10.Jx, 25.40.Lw, 97.10.Cv I. INTRODUCTION The rp process is the dominating nuclear process in hot and explosive hydrogen burning scenarios 1 4. The process is characterized by a large number of proton capture reactions and decays. It is responsible for the energy production and nucleosynthesis in a variety of astrophysical sites with different temperature and density conditions like novae, x-ray bursts, x-ray pulsars, and maybe accretion disks around low mass black holes. Ever since the rp process was introduced by Wallace and Woosley in 1981 the very special role of the doubly magic nucleus 56 Ni has been recognized. At the high temperature and density conditions in x-ray bursts and x-ray pulsars 56 Ni is a so called waiting point in the rp-process path. Waiting points are isotopes with a small or negative Q value for proton capture. In these cases proton capture is hampered by either photodisintegration or proton decay and the reaction flow has usually to wait for the relatively slow decay. For 56 Ni the low proton capture Q value of 695 kev coincides with a low proton capture rate owing to the low level density near the closed shell and the extraordinary long -decay lifetime. The long terrestrial half-life of 6.075 d 5 is considerably prolonged for the fully ionized atom at stellar conditions since the decay is dominated by electron capture 6,7. In fact, 56 Ni is the only waiting point nucleus in the rp process which has a -decay lifetime exceeding all typical processing time scales. Therefore, at 56 Ni the rp process either ends or has to proceed via proton captures. Most of the earlier studies assumed that the rp process ends at 56 Ni 1,8 10 or that at least processing beyond 56 Ni is negligible 11. However, more recent studies based on larger reaction networks show that the rp process in x-ray bursts and x-ray pulsars proceeds well beyond 56 Ni 12 14 and can even reach the A80100 mass region 15 18. This has far *Present address: CERN, CH-1211 Genève 23, Switzerland. Present address: Department of Physics and Astronomy, and National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI 48824. reaching consequences for the consumption of hydrogen and helium fuel as well as for the energy production and the final composition of the rp process ashes. This in turn is linked to a variety of neutron star properties and observables see Schatz et al. 17 and references therein. However, these conclusions depend sensitively on the proton capture rates on 56 Ni and, when at high temperatures 56 Ni and 57 Cu are in (p,)-(,p) equilibrium, on 57 Cu. For the 56 Ni(p,) 57 Cu reaction several calculations have been performed in the past. A more recent estimate was made by Van Wormer et al. 13, but at that time only incomplete information about the level structure in 57 Cu was available. The situation improved considerably after Zhou et al. 19 remeasured the level structure of 57 Cu and evaluated an updated reaction rate. A first attempt to measure the spectroscopic factors of the low lying proton unbound states in 57 Cu was made by Rehm et al. 20 who populated the analogue states using the d( 56 Ni, 57 Ni)p reaction in a radioactive beam experiment. Rehm et al. also presented a new shell model calculation for the 56 Ni(p,) 57 Cu reaction rate and found, within the relatively large errors of the model dependent analysis, agreement between their calculated and measured spectroscopic factors. Although the spectroscopic factors are in agreement with Zhou et al. 19, the derived reaction rates differ substantially. For the 57 Cu(p,) 58 Zn reaction only a rate calculated with the statistical Hauser-Feshbach model is available 21. However, this approach might be doubtful as in this case the level density in the compound nucleus is quite small. We present a new analysis of the reactions 56 Ni(p,) 57 Cu and 57 Cu(p,) 58 Zn based on improved theoretical calculations combined with all available experimental data. In the shell model we used the most recent interaction from Ormand 1997 22. Proton widths were calculated using a folding potential. For experimentally unknown excitation energies and spin-parity assignments we calculated Coulomb and Thomas-Ehrman shifts to obtain the energies starting from the experimentally known levels in the mirror nuclei. Spectroscopic factors were calculated using the nuclear shell model. Gamma widths were obtained from the experimental information in the mirror nuclei. In Sec. II we describe briefly the formalism for calculat- 0556-2813/2001/644/0458019/$20.00 64 045801-1 2001 The American Physical Society

FORSTNER, HERNDL, OBERHUMMER, SCHATZ, AND BROWN PHYSICAL REVIEW C 64 045801 ing stellar reaction rates. The experimental and theoretical nuclear input parameters for the calculation of the reaction rates are presented in Sec. III. Results and astrophysical implications are discussed in Sec. IV. A summary is given in Sec. V. II. CALCULATION OF STELLAR REACTION RATES The proton capture cross sections on the investigated nuclei are predominantly determined by summing the contributions from isolated resonances corresponding to unbound compound nuclear states and from the nonresonant direct capture DC process. In the following we briefly describe the method of calculating the resonant and nonresonant DC contributions to the stellar reaction rates. A. Resonant reaction contributions For the reactions considered in this work the resonances are narrow and isolated. The resonant rate contribution can be calculated from resonance energies E i and resonance strengths () i both in units of MeV 23 N A v r 1.5410 11 T 9 3/2 i i exp11.605e i /T 9 cm 3 mol 1 s 1. The resonance strength of an isolated resonance for a (p,) reaction is given by 2J1 22 j t 1 1 p tot, 2 where J and j t are the spins of the resonance and the target nucleus, respectively, and the total width tot is the sum of the proton width p and the gamma width. The proton width p can be estimated from the singleparticle spectroscopic factor S and the single-particle width sp of the resonance by using 24 p C 2 S sp, 3 FIG. 1. Double folding potential used for the reaction 56 Ni(p,) 57 Cu dashed line and for 57 Cu(p,) 58 Zn dotted line. where C is the isospin Clebsch-Gordan coefficient. The spectroscopic factors S are calculated in this work by using the nuclear shell model Sec. III. Single-particle widths sp are obtained from resonant scattering phase shifts generated by an appropriate folding potential see below. In this context, the partial width sp is defined as the energy interval over which the resonant phase shift varies from /4 to 3/4. Gamma widths for specific electromagnetic transitions are expressed in terms of reduced transition probabilities B(J i J f ;L) which contain the necessary nuclear structure information of the states involved in the transition 25. The reduced transition probabilities can be calculated in the framework of the nuclear shell model. Alternatively they can be deduced from the experimentally known lifetimes of the corresponding states in the mirror nuclei. This method is used in this work Sec. III. The total gamma width of a particular resonance is given by the sum over partial gamma widths for transitions to all possible lower-lying nuclear states. B. Nonresonant reaction contributions The nonresonant proton capture cross section is calculated by using the direct capture DC model described in 26 28. The DC cross section DC i for a particular transition is determined by the overlap of the scattering wave function in the entrance channel, the bound-state wave function in the exit channel and the electromagnetic multipole transition operator. Usually, only the dominant E1 transitions have to be taken into account. Wave functions are obtained by using a real folding potential given by 27,29 VRV F R a r 1 A r 2 v eff E, a, A,sdr 1 dr 2. Here represents a potential strength parameter close to unity, and srr 2 r 1, with R the separation of the centers of mass of the projectile and the target nucleus. Since there exists no measured charge distributions 30 for 56 Ni and 57 Cu we used densities obtained using the Hartree-Fock method as described in Ref. 31. The effective nucleonnucleon interaction v eff has been taken in the DDM3Y parametrization 29. The imaginary part of the potential has been neglected due to the small flux into other reaction channels. The obtained real folding potential for the two reactions is shown in Fig. 1. The total nonresonant cross section nr is determined by summing contributions of direct capture transitions to all bound states with single-particle spectroscopic factors S i : 4 nr i C 2 S i i DC. 5 045801-2

THERMONUCLEAR REACTION RATE OF... PHYSICAL REVIEW C 64 045801 TABLE I. Coulomb shifts E C, Thomas-Ehrman shifts E TE, and excitation energies E x calculated for 57 Cu and 58 Zn. Excitation energies are given in MeV. Explanation in the text. J E x ( 57 Ni) E C E TE E x ( 57 Cu) 5/2 0.77 0.061 0.16 0.99 J E x ( 58 Ni) E C E mathte E x ( 58 Zn) 2 1 4 1 2 2 1 1 0 2 2 3 2 4 3 1 0 3 1 2 3 2 1.45 0.051 0.009 1.39 2.46 0.010 0.11 2.55 2.78 0.078 0.001 2.70 2.90 0.032 0.11 2.97 2.94 0.009 0.11 3.06 3.04 0.060 0.062 3.04 3.26 0.038 0.12 3.35 3.42 0.051 0.12 3.49 3.53 0.12 0.027 3.39 3.59 0.11 0.068 3.42 3.77 0.073 0.11 3.81 The astrophysical S factor of a charged-particle induced reaction is defined by 23 SEE exp2e, with denoting the Sommerfeld parameter. If the S factor depends only weakly on the bombarding energy the nonresonant reaction rate as a function of temperature T 9 can be expressed as 1/3 2 AT 9 N A v nr 7.83310 9 Z 1Z 2 exp 4.249 Z 1 2 Z 2 2 A T 9 SE 0 MeV b 6 1/3 cm 3 mol 1 s 1, with Z 1 and Z 2 the charges of the projectile and target, respectively, and A the reduced mass in amu. The quantity E 0 denotes the position of the Gamow peak corresponding to the effective bombarding energy range of stellar burning. TABLE II. Shell model spectroscopic factors of 57 Cu bound and unbound states. C 2 S E x MeV J 2p 1/2 2p 3/2 1 f 5/2 1 f 7/2 lp l f 0.000 3/2 0.91 1.028 5/2 0.91 0.99 1.106 1/2 0.90 1.03 2.398 5/2 0.028 a 0.95 2.520 7/2 0.01 a 0.95 a From Zhou et al. 19. 7 TABLE III. Shell model spectroscopic factors of 58 Zn bound and unbound states. C 2 S E x MeV J 2p 1/2 2p 3/2 1 f 5/2 lp l f 0.000 0 1.20 1.40 2 0.17 0.98 0.11 2.55 4 0.95 0.99 2.70 2 0.16 0.99 0.22 1.02 0.98 2.97 1 1.0 0.98 3.04 2 0.29 0.023 0.64 1.0 0.97 3.06 0 0.79 1.0 3.35 2 0.34 0.011 0.024 1.0 0.97 3.39 0 0.012 0.99 3.42 1 1.0 0.99 3.49 3 1.0 0.96 3.52 4 0.05 a 0.96 3.56 4 0.05 a 0.96 3.62 4 0.05 a 0.96 3.81 3 1.0 0.98 a Assumption. III. EXPERIMENTAL AND THEORETICAL INPUT PARAMETERS In this section we present a discussion of the excitation energies E x, spectroscopic factors S, proton widths p, gamma widths, and resonance strengths. We use the nuclear shell model to calculate excitation energies, Coulomb shifts and spectroscopic factors. The code OXBASH 32 has been employed for this purpose. We use the isospin nonconserving INC interaction that has been developed by Ormand 22. It is especially designed to describe the protonrich isotopes in the mass range above 56 Ni. The relevant excitation energies in 57 Cu have been measured by Zhou et al. 19. For 58 Zn no experimental excitation energies are available. Therefore we use the known excitation energies in the mirror nucleus 58 Ni 33 and calculate Coulomb shifts in the shell model. In the mass region above the double magic nucleus 56 Ni considerable Coulomb shifts between analogue states of mirror nuclei exist. The situation is similar to the mass region above the doubly magic nucleus 16 O. For some levels the strong shifts cannot be explained with the Coulomb shifts resulting from an INC interaction. This fact has been explained by the Thomas-Ehrman shift 34,35. Since the proton single particle wave functions are only loosely bound, they are pushed out into the nuclear exterior. As a consequence the Coulomb energy is reduced. If two shell model orbits with different angular momenta are filled up different effects can occur. While in the region above 16 O the subshells 1d 5/2 and 2s 1/2 are filled up, in the region above 56 Ni a competition between the 2p 3/2 and the 1 f 5/2 subshell takes place. However, in both cases the subshell with the lower angular momentum is pushed out much more and therefore experiences a smaller Coulomb energy. We have used the method developed in Ref. 36 to cal- 045801-3

FORSTNER, HERNDL, OBERHUMMER, SCHATZ, AND BROWN PHYSICAL REVIEW C 64 045801 TABLE IV. Resonance parameters for the reaction 56 Ni(p,) 57 Cu. E x kev a J f E r kev b p ev ev ev 1028 5/2 333 5.4510 12 3.5510 4 1.6410 11 1106 1/2 411 1.7210 7 4.2310 3 1.7210 7 2398 5/2 1703 9.1310 2 1.3710 2 3.5710 2 2520 7/2 1825 7.0810 2 8.3810 3 3.0010 2 a Experimental values adopted from Ref. 33. b Calculated from column 1 and Q p 695 kev 33. culate the Thomas-Ehrman shifts for 57 Cu and 57 Ni as well as 58 Zn and 58 Ni. We have not taken into account possible complications due to the 2p 1/2 subshell. In general the excited states have a stronger component of the 1 f 5/2 -wave configuration than the ground state. Therefore the excited state of the proton-rich nucleus tends to be at a higher excitation energy than its analogue state. We have calculated the shifts for the excited states of 57 Cu and 58 Zn starting from their analogue states in 57 Ni and 58 Ni. The resulting Thomas-Ehrman shift E TE is then added to the Coulomb shift E C obtained from the shell model calculation with the INC interaction. The results are listed in Table I. We tested the method on the known first excited 5/2 state in 57 Cu. Starting from the excitation energy of 768 kev in 57 Ni we calculated an excitation energy of 991 kev in the mirror nucleus, which is only 37 kev off the measured energy of 1028 kev see Table I. This gives us confidence that we can also use this method for the calculation of the excitation energies of 58 Zn. Table I shows that all states experience relatively low shifts of not more than 170 kev. The shifts for the second, third, and fourth excited state in 57 Cu are not included in Table I. The second excited 1/2 state in 57 Cu is a good 2p 1/2 single-particle state see the discussion of the spectroscopic factors below. Since this subshell is not included in our model of the Thomas-Ehrman shift we did not calculate a shift for this state. The next two excited states (5/2 and 7/2 ) are poorly described by the interaction and model space we have used 22. However, the excited states of 58 Zn are well described in the model space which can be seen from the single particle spectroscopic factors. The calculated spectroscopic factors are listed in Tables II and III. For the first two excited states in 57 Cu our shell model calculations agree with the experimental results of Rehm et al. 20. For the third and fourth excited state of 57 Cu we use the spectroscopic factors given by Zhou et al. 19. It will be interesting to investigate these spectroscopic factors with the larger model space and new effective interactions which are being developed for the fp shell 37. In this work the gamma widths are deduced from the experimentally known lifetimes of the corresponding states in the mirror nuclei. We have corrected the transition energies using our adopted excitation energies and the well known energy dependence of the E1, M1, and E2 transitions. The obtained widths and resonance strengths are shown in Tables IV and V. IV. RESULTS AND DISCUSSION The resulting stellar rates of the 56 Ni(p,) 57 Cu and 57 Cu(p,) 58 Zn reaction can be parametrized for temperatures below T 9 2 by the expression 38 E x kev a J f TABLE V. Resonance parameters for the reaction 57 Cu(p,) 58 Zn. E R kev b p ev ev ev 2554 4 274 8.6510 15 5.6810 4 9.7310 15 2697 2 417 7.7110 8 1.1310 3 4.8210 8 2971 1 691 5.6510 6 8.4510 3 2.1210 6 3040 2 760 1.910 3 1.2110 2 1.0210 3 3057 0 777 6.7510 3 3.0310 6 3.7810 7 3349 2 1069 0.33 2.2010 2 1.2910 2 3389 0 1109 1.8810 2 1.9210 3 2.1810 4 3420 1 1140 2.22 1.1210 2 4.1710 3 3492 3 1212 3.1310 2 1.7010 3 1.4110 3 3524 4 1244 2.1610 3 6.8810 3 1.8510 3 3558 4 1278 3.2410 3 7.5210 3 2.5510 4 3620 4 1340 5.7710 3 3.3110 3 2.3710 9 3807 3 1527 4.4510 3 1.4510 3 9.5510 4 a Values taken from the last column of Table I. b Calculated from column 1 and Q p 2280 kev 33. 045801-4

THERMONUCLEAR REACTION RATE OF... PHYSICAL REVIEW C 64 045801 TABLE VI. Recommended parameters for the 56 Ni(p,) 57 Cu reaction rate. The total stellar reaction rate is given by Eq. 10. A i B i C D 2.5910 6 3.86 2.2110 9 39.3 2.7210 2 4.77 5.6510 3 19.8 4.7410 3 21.2 N A v i A i /T 9 3/2 expb i /T 9 C/T 9 2/3 expd/t 9 1/3 cm 3 mol 1 s 1. The first and second term in Eq. 8 represent the contributions of all narrow resonances and the direct capture process, respectively. The parameters A i, B i, C, and D are listed in Tables VI and VII. The various contributions to the total reaction rates are displayed in Figs. 2 and 3. For both reactions the direct capture process can be neglected in the temperature range from T 9 0.2 to T 9 2. The 56 Ni(p,) 57 Cu rate below 1 GK is clearly dominated by the resonant capture to the 1106 kev state. For higher temperatures the resonances of the 2398 kev and 2520 kev states determine the reaction rate. In the reaction 57 Cu(p,) 58 Zn the three 2 resonances dominate the reaction rate. We compare our 56 Ni(p,) 57 Cu rate with the previous results in Fig. 4. Below a temperature of 1 GK our rate agrees with Rehm et al. 20 while it is about one order of magnitude larger than the rate of Zhou et al. 19 and up to two orders of magnitude larger than the rate of Van Wormer et al. 13. In all calculations approximately the same spectroscopic factors Van Wormer et al.: 0.8; Zhou et al.: 0.76; Rehm et al.: 0.9; this work: 0.9 have been used for the dominating 1/2 resonance. The large discrepancies to the reaction rate of Zhou et al. is therefore only due to differences in the calculation of the penetrability. Our folding po- 8 tential method agrees with the Woods-Saxon and R-matrix calculations performed by Rehm et al., but gives a proton width one order of magnitude larger than the extended unified model EUM calculations of Zhou et al. Van Wormer et al. used another method to calculate the penetrability and different resonance energies. At higher temperatures about 1.2 GK the rates of Rehm et al. and Zhou et al. are almost identical because Rehm et al. used the resonance parameters from Zhou et al. for the 2398 and 2520 kev state. A slight difference remains as Rehm et al. made an additional approximation assuming p and consequently for these states. Our rate is in this temperature regime about a factor of 2 smaller due to the smaller widths of these two states. We used the available experimental information from the mirror states to deduce the widths while Zhou et al. calculated them in the EUM. In Fig. 5 our 57 Cu(p,) 58 Zn rate is compared with the Hauser-Feshbach calculations using the code SMOKER 39,40. The dashed line shows the old calculation based on a theoretical level density in 58 Zn and the dotted line shows a calculation using the new 58 Zn levels derived in the present work. According to Rauscher et al. 41 Hauser-Feshbach calculations for the 57 Cu(p,) 58 Zn reaction rate are only reliable for T 9 2.3, because at lower temperatures the level density in 58 Zn is too small. Indeed below 2 GK both Hauser-Feshbach calculations give reaction rates that are an order of magnitude too low. As expected the results from the Hauser-Feshbach calculations are getting closer to our results at temperatures near 2 GK. Above 2 GK our reaction rate is smaller than the Hauser-Feshbach calculations because we did not include higher levels in our shell-model calculations. These results clearly demonstrate that Hauser-Feshbach calculations cannot be used to calculate the 57 Cu(p,) 58 Zn reaction rate for most rp process temperatures. The criterion given by Rauscher et al. 41 seems to be reliable for the determination of the temperature range where the Hauser- Feshbach approach can be applied. The dominating remaining uncertainties are for the TABLE VII. Recommended parameters for the 57 Cu(p,) 58 Zn reaction rate. The total stellar reaction rate is given by Eq. 10. A i B i C D 1.5410 9 3.18 1.3910 9 40.3 7.6210 3 4.84 3.3510 1 8.02 1.6210 2 8.82 5.9810 2 9.02 2.0410 3 12.4 3.4510 1 12.9 6.6010 2 13.2 2.2310 2 14.1 2.9310 2 14.4 4.0310 2 14.9 3.7410 2 15.6 1.5110 2 17.7 FIG. 2. Total stellar rate solid line and individual contributions dashed lines for the reaction 56 Ni(p,) 57 Cu. 045801-5

FORSTNER, HERNDL, OBERHUMMER, SCHATZ, AND BROWN PHYSICAL REVIEW C 64 045801 FIG. 3. Total stellar rate solid line and individual contributions dashed lines for the reaction 57 Cu(p,) 58 Zn. 56 Ni(p,) 57 Cu reaction the spectroscopic factors of the 2398 and 2520 kev states in 57 Cu. In 58 Zn some experimental information on excitation energies and spectroscopic factors of the 2 states would reduce the uncertainties. A nonnegligible contribution to the rate uncertainties also comes from the experimental errors in the masses of 56 Ni 11 kev, 57 Cu 16 kev, and 58 Zn 50 kev 42. In Figs. 6 and 7 the Q-value dependence of the reaction rate is shown. The mass uncertainties have the strongest impact on the 56 Ni(p,) 57 Cu rate below 1 GK, where the resulting uncertainty is up to a factor of 4. On the other hand, for the 57 Cu(p,) 58 Zn rate the influence of mass uncertainties is very small. More precise mass measurements of 56 Ni and 57 Cu would be desirable. FIG. 5. Comparison of the present reaction rate for 57 Cu(p,) 58 Zn to a previous calculation with the code SMOKER 39,40. V. ASTROPHYSICAL IMPLICATIONS The new 56 Ni(p,) 57 Cu and 57 Cu(p,) 58 Zn reaction rates allow us now to recalculate the effective rp-process half-life of 56 Ni against proton and electron capture. This was done by solving the differential equations describing the abundance of 56 Ni as a function of time for a fixed temperature and density and calculating the time required to reduce the 56 Ni abundance to half of its initial value. This represents a small reaction network including all reactions that affect the 56 Ni lifetime, which are proton captures on 56 Ni and 57 Cu, (,p) photodisintegration of 58 Zn, 57 Cu, and 56 Ni, as well as continuum electron capture on 56 Ni which turns out to be negligible and decays of 57 Cu and 58 Zn. The reaction rates used besides the new proton capture reaction rates on 56 Ni and 57 Cu discussed in this work were described in detail in Van Wormer et al. 13. All calculations were per- FIG. 4. Comparison of the present reaction rate for 56 Ni(p,) 57 Cu to previous results of Zhou et al. 19 and Wormer et al. 13. FIG. 6. Dependence of the reaction rate for 56 Ni(p,) 57 Cu on the Q value. 045801-6

THERMONUCLEAR REACTION RATE OF... PHYSICAL REVIEW C 64 045801 FIG. 7. Dependence of the reaction rate for 57 Cu(p,) 58 Zn on the Q value. formed for solar hydrogen abundance. The result is shown in Fig. 8 as a function of temperature and at a fixed density of 10 6 g/cm 3, which is typical for x-ray bursts and x-ray pulsars. Screening has been taken into account according to Refs. 43,44. For comparison, Fig. 8 also shows the same calculation based on the frequently used rate of Van Wormer FIG. 8. Upper panel: Lifetime of 56 Ni: a 56 Ni(p,) 57 Cu rate taken from Van Wormer et al. 13 and 57 Cu(p,) 58 Zn rate taken from Hauser-Feshbach calculations 39,40; b new 56 Ni(p,) 57 Cu rate and 57 Cu(p,) 58 Zn rate taken from Hauser-Feshbach calculations 39,40; c new 56 Ni(p,) 57 Cu rate and new 57 Cu(p,) 58 Zn rate. Lower panel: 55 Ni bypass. et al. 13 and, to demonstrate the influence of the different reactions, also for the new 56 Ni(p,) 57 Cu reaction rate together with the old Hauser-Feshbach rate for 57 Cu(p,) 58 Zn. For temperatures between 0.35 and 0.95 GK the new 56 Ni(p,) 57 Cu rate leads to a reduction of the 56 Ni lifetime by 1 2 orders of magnitude. However, for the interpretation of the consequences for the rp process it has to be taken into account that there is a possibility of bypassing 56 Ni via the 55 Ni(p,) 56 Cu(p,) 57 Zn( ) 57 Cu reaction sequence for temperatures between 0.65 and 1.1 GK bypass 50% see the lower panel of Fig. 8. For this temperature range the 56 Ni lifetime has no strong impact on the rp process. However, the calculations for this bypass sequence are quite uncertain and based on the theoretical mass predictions of Ormand 22. An experimental determination of the 56 Cu mass would be very important. At higher temperatures the 56 Ni(p,) 57 Cu reaction is very fast, but (,p) photodisintegration of the relatively weakly proton-bound 57 Cu hampers effective proton captures leading again to higher lifetimes. Two cases have to be distinguished. 1 as long as photodisintegration on the more proton bound 58 Zn does not play a role, there is a (p,)-(,p) equilibrium established between 56 Ni and 57 Cu, but not between 57 Cu and 58 Zn. Then the lifetime of 56 Ni depends on the mass difference between 56 Ni and 57 Cu and the 57 Cu(p,) 58 Zn reaction rate. 2 However, above the temperature where photodisintegration on 58 Zn starts playing a role, 56 Ni, 57 Cu, and 58 Zn are in (p,)-(,p) equilibrium and then the 56 Ni lifetime depends not on the 57 Cu(p,) 58 Zn reaction rate anymore, but on the mass difference between 56 Ni and 58 Zn and the decay rate of 58 Zn. It turns out that because the new 57 Cu(p,) 58 Zn is so large, case 2 sets in at a lower temperature than case 1 despite of the fact that 58 Zn is more proton bound. This happens because as a consequence of the detailed balance principle a higher 57 Cu(p,) 58 Zn reaction rate means also a higher 58 Zn photodisintegration rate. Therefore, photodisintegration of 58 Zn sets already in at lower temperatures case 2, while at the same time photodisintegration of 57 Cu requires higher temperatures to compete with the increased proton capture rate case 1. Above a certain 57 Cu(p,) 58 Zn rate the temperature conditions for the onset of case 1 and 2 crossover, and case 1 is skipped. Therefore, with the new 57 Cu(p,) 58 Zn reaction rate it can be concluded that its not the 57 Cu(p,) 58 Zn reaction rate but the decay rate of 58 Zn that determines the 56 Ni lifetime against proton capture at high temperatures. This is illustrated in Fig. 8, which shows that there is hardly any difference in the 56 Ni lifetime calculated with the old and with the new 57 Cu(p,) 58 Zn reaction rate. The possible influence of the new 56 Ni lifetime against proton capture on the rp process in x-ray bursts and x-ray pulsars is discussed in the following. In x-ray bursts the important question is whether the rp process can proceed beyond 56 Ni within the typical burst time scale of 10 s. Figure 8 shows that there is indeed a temperature window around 1 GK, where the lifetime of 56 Ni is below 10 s. The new reaction rates enlarge this window considerably and it ranges now from 0.33 to 1.95 GK 045801-7

FORSTNER, HERNDL, OBERHUMMER, SCHATZ, AND BROWN PHYSICAL REVIEW C 64 045801 instead of 0.45 1.9 GK with the Van Wormer et al. 13 rates. Therefore the previously made conclusions remain unchanged that for typical burst peak temperatures of 1 2 GK the rp process can well proceed beyond 56 Ni. However, the time during cooling where nuclei above 56 Ni can be produced becomes significantly longer. This will be especially important for short bursts, where the fraction of material getting stuck at 56 Ni will be reduced. An important new aspect is that with the new reaction rates the temperature threshold for rp processing beyond 56 Ni coincides roughly with the temperature limit for the breakout of the CNO cycle. So any burst that will be hot enough for the rp process to occur will also be hot enough for processing beyond 56 Ni. For bursts with peak temperatures near 2 GK the rp process gets stuck in 56 Ni until temperatures cool down to reach again a 56 Ni lifetime of around 10 s. The temperature where this happens sets the temperature for the rp process beyond 56 Ni. As discussed above, it is now clear from the new 57 Cu(p,) 58 Zn reaction rate that this temperature is not determined by the 57 Cu(p,) 58 Zn reaction, but by the 58 Zn decay rate. In x-ray pulsars hydrogen and helium are burned in steady state and the rp process will end when all the hydrogen is consumed. An important question is how much 56 Ni is produced in these scenarios. Older studies assumed that 56 Ni is the sole product of the nucleosynthesis and all theoretical descriptions used this as a starting point. Schatz et al. 17,18 showed that this is not the case and that depending on the accretion rate a variety of nuclei between A50 100 are made. However, they still found an appreciable amount of 56 Ni in the final composition of the rp-process ashes. This is due to the long lifetime of the order of seconds and more for 56 Ni for temperatures below 1 GK, which make 56 Ni a waiting point. The abundance accumulated at a waiting point is roughly proportional to its lifetime. Therefore, with the new reaction rates, the final 56 Ni abundance will be reduced by a factor of 10 100 see Fig. 8. With our new rates the conclusion will most likely be that there is no 56 Ni produced at all in x-ray pulsars. Finally, we give in Fig. 9 the lifetime of 56 Ni against proton capture as a function of temperature and density based on the new reaction rates. This plot can be read as the processing time that is needed for the rp process to get beyond 56 Ni for a given temperature and density condition. Of course this is a lower limit and the time for the rp process to reach 56 Ni has to be added. FIG. 9. Lifetime of 56 Ni in the rp process as a function of temperature and density. The contour level labels indicate the log 10 of the lifetime in seconds. The 56 Ni lifetime is mainly determined by proton captures, and a solar hydrogen abundance has been assumed. The energy shift of the first excited 5/2 state in 57 Cu has been calculated from the corresponding state of 57 Ni with the help of the Coulomb and Thomas-Ehrmann shifts. Excellent agreement with the experimentally obtained energy shift is obtained. In an analogous way using again the Coulomb and Thomas-Ehrmann shifts the excitation energies and spin/ parity assignments of the experimentally unknown levels of 58 Zn were obtained. The proton and gamma widths of the astrophysically relevant resonances are estimated with the help of shell model calculations. With these consistent experimental and theoretical input parameters the reaction rates for 56 Ni(p,) 57 Cu and 57 Cu(p,) 58 Zn have been calculated. The results confirm the new 56 Ni(p,) 57 Cu reaction rate given recently by Rehm et al. 20 for temperatures below 1 GK and lead to a significantly improved rate at higher temperatures. Compared with reaction rates previously used in astrophysical network calculations, this new rate drastically changes the lifetime of 56 Ni against proton capture in the rp process and reduces the minimum temperature required for the rp process to proceed beyond 56 Ni. For x-ray burst densities this minimum temperature coincides now with the temperature for the break out of the hot CNO cycles. We also present the first reliable calculation of the 57 Cu(p,) 58 Zn reaction rate. Based on this new rate we can now conclude that it is in fact not the 57 Cu(p,) 58 Zn reaction rate but the 58 Zn decay rate that determines the 56 Ni lifetime in the rp process at high temperatures. VI. SUMMARY AND CONCLUSIONS ACKNOWLEDGMENTS We would like to thank T. Rauscher for providing the latest Hauser-Feshbach calculations for the 57 Cu(p,) 58 Zn reaction rate and F.-K. Thielemann for supplying the reaction network code used for the calculation of the 56 Ni lifetime. This work was supported by Fonds zur Förderung der Wissenschaftlichen Forschung in Österreich Project No. S7307-AST. B.A.B. is supported by NSF Grant No. PHY- 9605207. 045801-8

THERMONUCLEAR REACTION RATE OF... PHYSICAL REVIEW C 64 045801 1 R. K. Wallace and S. E. Woosley, Astrophys. J., Suppl. Ser. 45, 389 1981. 2 M. Politano, S. Starrfield, J. W. Truran, A. Weiss, and W. M. Sparks, Astrophys. J. 448, 807 1995. 3 R. E. Taam, S. E. Woosley, and D. Q. Lamb, Astrophys. J. 459, 271 1996. 4 L. Jin, W. D. Arnett, and S. K. Chakrabarti, Astrophys. J. 336, 572 1989. 5 Karlsruher Nuklidkarte, 6. Auflage 1995. 6 B. Sur, E. B. Norman, K. T. Lesko, E. Browne, and R.-M. Larimer, Phys. Rev. C 42, 573 1990. 7 G. M. Fuller, W. A. Fowler, and M. J. Newman, Astrophys. J., Suppl. 42, 447 1980. 8 S. E. Woosley and T. A. Weaver, in High Energy Transients in Astrophysics, edited by S. E. Woosley, AIP Conf. Proc. No. 115 American Institute of Physics, New York, 1984, p. 273. 9 R. E. Taam, S. E. Woosley, T. A. Weaver, and D. Q. Lamb, Astrophys. J. 413, 324 1993. 10 R. E. Taam, S. E. Woosley, and D. Q. Lamb, Astrophys. J. 459, 271 1996. 11 R. K. Wallace and S. E. Woosley, in High Energy Transients in Astrophysics Ref. 8, p. 319. 12 T. Hanawa, D. Sugimoto, and M. A. Hashimoto, Publ. Astron. Soc. Jpn. 35, 491 1983. 13 L. van Wormer, J. Görres, C. Iliadis, M. Wiescher, and F.-K. Thielemann, Astrophys. J. 432, 326 1994. 14 O. Koike, M. Hashimoto, K. Arai, and S. Wanajo, Astron. Astrophys. 342, 464 1999. 15 Intersections between Particle and Nuclear Physics, edited by H. Schatz, L. Bildsten, J. Görres, F.-K. Thielemann, M. Wiescher, and T. W. Donnelly, 6th Conference, Big Sky, Montana, AIP Conf. Proc. No. 412 American Institute of Physics, New York, 1997, p. 987. 16 H. Schatz, A. Aprahamian, J. Görres, M. Wiescher, T. Rauscher, J. F. Rembges, F.-K. Thielemann, B. Pfeiffer, P. Möller, K.-L. Kratz, H. Herndl, B. A. Brown, and H. Rebel, Phys. Rep. 294, 11998. 17 H. Schatz, L. Bildsten, A. Cumming, and M. Wiescher, in Nuclei in the Cosmos V, edited by N. Prantzos, Volos, Greece, 1998 in press. 18 H. Schatz, L. Bildsten, A. Cumming, and M. Wiescher, Astrophys. J. 524, 1014 1999. 19 X. G. Zhou et al., Phys. Rev. C 53, 982 1996. 20 K. E. Rehm et al., Phys. Rev. Lett. 80, 676 1998. 21 F.-K. Thielemann private communication. 22 W. E. Ormand, Phys. Rev. C 55, 2407 1997. 23 C. E. Rolfs and W. S. Rodney, Cauldrons in the Cosmos The University of Chicago Press, Chicago, 1988. 24 J. P. Schiffer, Nucl. Phys. 46, 246 1963. 25 P. J. Brussaard and P. W. M. Glaudemans, Shell-Model Applications in Nuclear Spectroscopy North-Holland, Amsterdam, 1977. 26 K. H. Kim, M. H. Park, and B. T. Kim, Phys. Rev. C 35, 363 1987. 27 H. Oberhummer and G. Staudt, in Nuclei in the Cosmos, edited by H. Oberhummer Springer-Verlag, Berlin, New York, 1991, p.29. 28 P. Mohr, H. Abele, R. Zwiebel, G. Staudt, H. Krauss, H. Oberhummer, A. Denker, J. W. Hammer, and G. Wolf, Phys. Rev. C 48, 1420 1993. 29 A. M. Kobos, B. A. Brown, R. Lindsay, and G. R. Satchler, Nucl. Phys. A425, 205 1984. 30 H. de Vries, C. W. de Jager, and C. de Vries, At. Data Nucl. Data Tables 36, 495 1987. 31 J. Street, B. A. Brown, and P. E. Hodgson, J. Phys. G 8, 839 1982. 32 B. A. Brown, A. Etchegoyen, W. D. M. Rae, and N. S. Godwin, code OXBASH, 1984 unpublished. 33 R. B. Firestone, Table of Isotopes, 8th ed. Wiley, New York, 1996. 34 R. F. Thomas, Phys. Rev. 81, 148 1951. 35 J. B. Ehrman, Phys. Rev. 81, 412 1951. 36 H. Herndl, J. Görres, M. Wiescher, B. A. Brown, and L. van Wormer, Phys. Rev. C 52, 1078 1995. 37 T. Mizusaki, T. Otsuka, Y. Utsuno, M. Honma, and T. Sebe, Phys. Rev. C 59, 1846 1999; E. Caurier, G. Martinez-Pinedo, F. Nowacki, A. Poves, J. Retamosa, and A. P. Zuker, ibid. 59, 2033 1999. 38 M. Wiescher, J. Görres, F.-K. Thielemann, and H. Ritter, Astron. Astrophys. 160, 561986. 39 F.-K. Thielemann et al., Advances in Nuclear Astrophysics Editions Frontiéres, Gif-sur-Yvette, 1987, p. 525. 40 J. J. Cowan, F.-K. Thielemann, and J. W. Truran, Phys. Rep. 208, 268 1991. 41 T. Rauscher, F.-K. Thielemann, and K.-L. Kratz, Phys. Rev. C 56, 1613 1997. 42 G. Audi and A. H. Wapstra, Nucl. Phys. A595, 409 1995. 43 H. C. Graboske, H. E. DeWitt, A. S. Grossman, and M. S. Cooper, Astrophys. J. 181, 457 1973. 44 N. Itoh, H. Totsuji, S. Ichimaru, and H. E. DeWitt, Astrophys. J. 234, 1079 1979. 045801-9